Geometric Series: Partial Sums in the Complex Plane

Lesson 3 · watch ∑ zⁿ converge — or refuse to — one partial sum at a time

Interactive: drag a point z in the complex plane and slide the number of terms N; the partial sums of the geometric series are drawn as a connected path that spirals into 1/(1−z) when |z| is less than 1.

Drag the white dot z. The path connects the partial sums S₀ = 1, S₁ = 1 + z, S₂ = 1 + z + z², … — each new segment is the next term zⁿ, so the path rotates by arg z and scales by |z| at every step. Inside the dashed unit circle the spiral closes in on 1/(1−z) (the × mark); outside, it flies apart; right on the circle it circles forever without settling. The amber ε-disk is the "challenge" of the ε–N definition of convergence: N(ε) is the cutoff past which every partial sum stays inside the disk — shrink ε and watch the required N grow. Points that are inside for good turn green.